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<div class="subTitle">org.apache.commons.math3.util</div>
<h2 title="Class CombinatoricsUtils" class="title">Class CombinatoricsUtils</h2>
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<pre>public final class <span class="strong">CombinatoricsUtils</span>
extends <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre>
<div class="block">Combinatorial utilities.</div>
<dl><dt><span class="strong">Since:</span></dt>
  <dd>3.3</dd></dl>
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<h3>Method Summary</h3>
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<caption><span>Methods</span><span class="tabEnd">&nbsp;</span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static long</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/CombinatoricsUtils.html#binomialCoefficient(int,%20int)">binomialCoefficient</a></strong>(int&nbsp;n,
                   int&nbsp;k)</code>
<div class="block">Returns an exact representation of the <a
 href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
 Coefficient</a>, "<code>n choose k</code>", the number of
 <code>k</code>-element subsets that can be selected from an
 <code>n</code>-element set.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/CombinatoricsUtils.html#binomialCoefficientDouble(int,%20int)">binomialCoefficientDouble</a></strong>(int&nbsp;n,
                         int&nbsp;k)</code>
<div class="block">Returns a <code>double</code> representation of the <a
 href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
 Coefficient</a>, "<code>n choose k</code>", the number of
 <code>k</code>-element subsets that can be selected from an
 <code>n</code>-element set.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/CombinatoricsUtils.html#binomialCoefficientLog(int,%20int)">binomialCoefficientLog</a></strong>(int&nbsp;n,
                      int&nbsp;k)</code>
<div class="block">Returns the natural <code>log</code> of the <a
 href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
 Coefficient</a>, "<code>n choose k</code>", the number of
 <code>k</code>-element subsets that can be selected from an
 <code>n</code>-element set.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static void</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/CombinatoricsUtils.html#checkBinomial(int,%20int)">checkBinomial</a></strong>(int&nbsp;n,
             int&nbsp;k)</code>
<div class="block">Check binomial preconditions.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static <a href="http://docs.oracle.com/javase/7/docs/api/java/util/Iterator.html?is-external=true" title="class or interface in java.util">Iterator</a>&lt;int[]&gt;</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/CombinatoricsUtils.html#combinationsIterator(int,%20int)">combinationsIterator</a></strong>(int&nbsp;n,
                    int&nbsp;k)</code>
<div class="block">Returns an iterator whose range is the k-element subsets of {0, ..., n - 1}
 represented as <code>int[]</code> arrays.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static long</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/CombinatoricsUtils.html#factorial(int)">factorial</a></strong>(int&nbsp;n)</code>
<div class="block">Returns n!.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/CombinatoricsUtils.html#factorialDouble(int)">factorialDouble</a></strong>(int&nbsp;n)</code>
<div class="block">Compute n!, the<a href="http://mathworld.wolfram.com/Factorial.html">
 factorial</a> of <code>n</code> (the product of the numbers 1 to n), as a
 <code>double</code>.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/CombinatoricsUtils.html#factorialLog(int)">factorialLog</a></strong>(int&nbsp;n)</code>
<div class="block">Compute the natural logarithm of the factorial of <code>n</code>.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static long</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/CombinatoricsUtils.html#stirlingS2(int,%20int)">stirlingS2</a></strong>(int&nbsp;n,
          int&nbsp;k)</code>
<div class="block">Returns the <a
 href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">
 Stirling number of the second kind</a>, "<code>S(n,k)</code>", the number of
 ways of partitioning an <code>n</code>-element set into <code>k</code> non-empty
 subsets.</div>
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<code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#clone()" title="class or interface in java.lang">clone</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#equals(java.lang.Object)" title="class or interface in java.lang">equals</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#finalize()" title="class or interface in java.lang">finalize</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#getClass()" title="class or interface in java.lang">getClass</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#hashCode()" title="class or interface in java.lang">hashCode</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notify()" title="class or interface in java.lang">notify</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notifyAll()" title="class or interface in java.lang">notifyAll</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#toString()" title="class or interface in java.lang">toString</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait()" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long)" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long,%20int)" title="class or interface in java.lang">wait</a></code></li>
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<h4>binomialCoefficient</h4>
<pre>public static&nbsp;long&nbsp;binomialCoefficient(int&nbsp;n,
                       int&nbsp;k)
                                throws <a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a>,
                                       <a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a>,
                                       <a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></pre>
<div class="block">Returns an exact representation of the <a
 href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
 Coefficient</a>, "<code>n choose k</code>", the number of
 <code>k</code>-element subsets that can be selected from an
 <code>n</code>-element set.
 <p>
 <Strong>Preconditions</strong>:
 <ul>
 <li> <code>0 &lt;= k &lt;= n </code> (otherwise
 <code>MathIllegalArgumentException</code> is thrown)</li>
 <li> The result is small enough to fit into a <code>long</code>. The
 largest value of <code>n</code> for which all coefficients are
 <code>&lt; Long.MAX_VALUE</code> is 66. If the computed value exceeds
 <code>Long.MAX_VALUE</code> a <code>MathArithMeticException</code> is
 thrown.</li>
 </ul></p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>n</code> - the size of the set</dd><dd><code>k</code> - the size of the subsets to be counted</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>n choose k</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>n &lt; 0</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a></code> - if <code>k &gt; n</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if the result is too large to be
 represented by a long integer.</dd></dl>
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<pre>public static&nbsp;double&nbsp;binomialCoefficientDouble(int&nbsp;n,
                               int&nbsp;k)
                                        throws <a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a>,
                                               <a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a>,
                                               <a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></pre>
<div class="block">Returns a <code>double</code> representation of the <a
 href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
 Coefficient</a>, "<code>n choose k</code>", the number of
 <code>k</code>-element subsets that can be selected from an
 <code>n</code>-element set.
 <p>
 <Strong>Preconditions</strong>:
 <ul>
 <li> <code>0 &lt;= k &lt;= n </code> (otherwise
 <code>IllegalArgumentException</code> is thrown)</li>
 <li> The result is small enough to fit into a <code>double</code>. The
 largest value of <code>n</code> for which all coefficients are less than
 Double.MAX_VALUE is 1029. If the computed value exceeds Double.MAX_VALUE,
 Double.POSITIVE_INFINITY is returned</li>
 </ul></p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>n</code> - the size of the set</dd><dd><code>k</code> - the size of the subsets to be counted</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>n choose k</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>n &lt; 0</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a></code> - if <code>k &gt; n</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if the result is too large to be
 represented by a long integer.</dd></dl>
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<pre>public static&nbsp;double&nbsp;binomialCoefficientLog(int&nbsp;n,
                            int&nbsp;k)
                                     throws <a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a>,
                                            <a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a>,
                                            <a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></pre>
<div class="block">Returns the natural <code>log</code> of the <a
 href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
 Coefficient</a>, "<code>n choose k</code>", the number of
 <code>k</code>-element subsets that can be selected from an
 <code>n</code>-element set.
 <p>
 <Strong>Preconditions</strong>:
 <ul>
 <li> <code>0 &lt;= k &lt;= n </code> (otherwise
 <code>MathIllegalArgumentException</code> is thrown)</li>
 </ul></p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>n</code> - the size of the set</dd><dd><code>k</code> - the size of the subsets to be counted</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>n choose k</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>n &lt; 0</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a></code> - if <code>k &gt; n</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if the result is too large to be
 represented by a long integer.</dd></dl>
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<h4>factorial</h4>
<pre>public static&nbsp;long&nbsp;factorial(int&nbsp;n)
                      throws <a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a>,
                             <a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></pre>
<div class="block">Returns n!. Shorthand for <code>n</code> <a
 href="http://mathworld.wolfram.com/Factorial.html"> Factorial</a>, the
 product of the numbers <code>1,...,n</code>.
 <p>
 <Strong>Preconditions</strong>:
 <ul>
 <li> <code>n &gt;= 0</code> (otherwise
 <code>MathIllegalArgumentException</code> is thrown)</li>
 <li> The result is small enough to fit into a <code>long</code>. The
 largest value of <code>n</code> for which <code>n!</code> does not exceed
 Long.MAX_VALUE} is 20. If the computed value exceeds <code>Long.MAX_VALUE</code>
 an <code>MathArithMeticException </code> is thrown.</li>
 </ul>
 </p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>n</code> - argument</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>n!</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if the result is too large to be represented
 by a <code>long</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>n &lt; 0</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if <code>n &gt; 20</code>: The factorial value is too
 large to fit in a <code>long</code>.</dd></dl>
</li>
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<pre>public static&nbsp;double&nbsp;factorialDouble(int&nbsp;n)
                              throws <a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></pre>
<div class="block">Compute n!, the<a href="http://mathworld.wolfram.com/Factorial.html">
 factorial</a> of <code>n</code> (the product of the numbers 1 to n), as a
 <code>double</code>.
 The result should be small enough to fit into a <code>double</code>: The
 largest <code>n</code> for which <code>n!</code> does not exceed
 <code>Double.MAX_VALUE</code> is 170. If the computed value exceeds
 <code>Double.MAX_VALUE</code>, <code>Double.POSITIVE_INFINITY</code> is returned.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>n</code> - Argument.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>n!</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>n &lt; 0</code>.</dd></dl>
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<pre>public static&nbsp;double&nbsp;factorialLog(int&nbsp;n)
                           throws <a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></pre>
<div class="block">Compute the natural logarithm of the factorial of <code>n</code>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>n</code> - Argument.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>n!</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>n &lt; 0</code>.</dd></dl>
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<pre>public static&nbsp;long&nbsp;stirlingS2(int&nbsp;n,
              int&nbsp;k)
                       throws <a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a>,
                              <a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a>,
                              <a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></pre>
<div class="block">Returns the <a
 href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">
 Stirling number of the second kind</a>, "<code>S(n,k)</code>", the number of
 ways of partitioning an <code>n</code>-element set into <code>k</code> non-empty
 subsets.
 <p>
 The preconditions are <code>0 &lt;= k &lt;= n </code> (otherwise
 <code>NotPositiveException</code> is thrown)
 </p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>n</code> - the size of the set</dd><dd><code>k</code> - the number of non-empty subsets</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>S(n,k)</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>k &lt; 0</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a></code> - if <code>k &gt; n</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if some overflow happens, typically for n exceeding 25 and
 k between 20 and n-2 (S(n,n-1) is handled specifically and does not overflow)</dd><dt><span class="strong">Since:</span></dt>
  <dd>3.1</dd></dl>
</li>
</ul>
<a name="combinationsIterator(int, int)">
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<h4>combinationsIterator</h4>
<pre>public static&nbsp;<a href="http://docs.oracle.com/javase/7/docs/api/java/util/Iterator.html?is-external=true" title="class or interface in java.util">Iterator</a>&lt;int[]&gt;&nbsp;combinationsIterator(int&nbsp;n,
                                   int&nbsp;k)</pre>
<div class="block">Returns an iterator whose range is the k-element subsets of {0, ..., n - 1}
 represented as <code>int[]</code> arrays.
 <p>
 The arrays returned by the iterator are sorted in descending order and
 they are visited in lexicographic order with significance from right to
 left. For example, combinationsIterator(4, 2) returns an Iterator that
 will generate the following sequence of arrays on successive calls to
 <code>next()</code>:</p><p>
 <code>[0, 1], [0, 2], [1, 2], [0, 3], [1, 3], [2, 3]</code>
 </p><p>
 If <code>k == 0</code> an Iterator containing an empty array is returned and
 if <code>k == n</code> an Iterator containing [0, ..., n -1] is returned.</p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>n</code> - Size of the set from which subsets are selected.</dd><dd><code>k</code> - Size of the subsets to be enumerated.</dd>
<dt><span class="strong">Returns:</span></dt><dd>an <a href="http://docs.oracle.com/javase/7/docs/api/java/util/Iterator.html?is-external=true" title="class or interface in java.util"><code>iterator</code></a> over the k-sets in n.</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>n &lt; 0</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a></code> - if <code>k &gt; n</code>.</dd></dl>
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<h4>checkBinomial</h4>
<pre>public static&nbsp;void&nbsp;checkBinomial(int&nbsp;n,
                 int&nbsp;k)
                          throws <a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a>,
                                 <a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></pre>
<div class="block">Check binomial preconditions.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>n</code> - Size of the set.</dd><dd><code>k</code> - Size of the subsets to be counted.</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>n &lt; 0</code>.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a></code> - if <code>k &gt; n</code>.</dd></dl>
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